The existence of the centre-mode linear instability at \(Re = 0\)  is already known in the limit of very dilute polymer solutions (\(\beta \rightarrow 1\) ) for \(Wi = O (10^3)\)  in Oldroyd-B fluids [1]} and for \(Wi = O (10^2)\)  in FENE-P fluids at finite extensibility (\(L_{max}\) ) [2]}. To substantiate its connection to ET, the time evolution of these growing modes has to be tracked to see whether they are able to produce turbulent behaviour, presumably after
transitioning through a cascade of intermediate states. Our goal here is to see where the first level of intermediate state - the TWs - exist at low and vanishing \(Re\) .
